The content introduces Ptolemy's Theorem, a generalization of the Pythagorean theorem, stating that for any cyclic quadrilateral, the sum of the products of opposite sides equals the product of its diagonals.
We all know that for any right triangle, A squared plus B squared equals C squared. But check it out:
A squared is A times A. B squared is B times B. And C squared is C times C. So
we could also say that for any rectangle, if you multiply the lengths of the opposite sides,
and add those two products together, you get the product of the two diagonals. I know,
mind blown right? But hear me out. If we give each line segment its own label, we can ask: is this
more general formula true for any quadrilateral? Well, no, it's not, but it is true for this one...
and this one... and this one. What is it that these quadrilaterals have in common with
rectangles that makes this formula work? The answer is they are all cyclic quadrilaterals,
which means that there is a circle that goes through each of the four vertices. If that's the
case then the sum of the products of the opposite sides will equal the product of the two diagonals.
This is Ptolemy's Theorem. So, what is it about a cyclic quadrilateral that makes this work?
Well, notice that these two angles are the same because of the inscribed angle theorem
which says that lines drawn from two points on a circle to a third point on the arc
between them will form an angle that is the same for all the points on that arc.
And given that, we can prove Ptolemy's Theorem. I'll let the animation take it from here...
[Music]
[Theme Song] MathyJaphy is here to say, let's do some math today. Today!
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